If \(X_0\) is the path-component of a space \(X\) containing the basepoint \(x_0\), show that the inclusion \(i: X_0 \hookrightarrow X\) induces an isomorphism \(i_*: \pi_1(X_0, x_0) \to \pi_1(X, x_0)\).
Solution
Goal: Prove that the inclusion \(i: X_0 \hookrightarrow X\) induces an isomorphism \(i_*: \pi_1(X_0, x_0) \to \pi_1(X, x_0)\) on fundamental groups.
1. Key Lemma: Any continuous map from a path-connected space into \(X\) sending a basepoint to \(x_0\) has image contained in \(X_0\).Proof:
1.1. Let \(Y\) be a path-connected space, \(y_0 \in Y\), and let \(f: Y \to X\) be a continuous map with \(f(y_0) = x_0\).
1.2. For any point \(y \in Y\), there exists a continuous path \(\gamma: [0, 1] \to Y\) with \(\gamma(0) = y_0\) and \(\gamma(1) = y\).
1.3. The composite \(f \circ \gamma: [0, 1] \to X\) is a continuous path in \(X\) starting at \((f \circ \gamma)(0) = f(y_0) = x_0\) and ending at \((f \circ \gamma)(1) = f(y)\).
1.4. Since \(X_0\) is the path-component of \(x_0\), by definition \(X_0\) contains all points in \(X\) path-connected to \(x_0\).
1.5. Therefore \(f(y) \in X_0\) for every \(y \in Y\), so \(f(Y) \subseteq X_0\).
2. Surjectivity of \(i_*: \pi_1(X_0, x_0) \to \pi_1(X, x_0)\):Proof:
2.1. Let \([\gamma] \in \pi_1(X, x_0)\) be represented by a continuous loop \(\gamma: [0, 1] \to X\) with \(\gamma(0) = \gamma(1) = x_0\).
2.2. Since the unit interval \([0, 1]\) is path-connected and \(\gamma(0) = x_0\), 1 implies \(\gamma([0, 1]) \subseteq X_0\).
2.3. Thus \(\gamma\) restricts to a well-defined continuous loop \(\gamma_0: [0, 1] \to X_0\) based at \(x_0\).
2.4. The homotopy class \([\gamma_0] \in \pi_1(X_0, x_0)\) satisfies \(i_*([\gamma_0]) = [i \circ \gamma_0] = [\gamma]\).
2.5. Therefore \(i_*\) is surjective.
3. Injectivity of \(i_*: \pi_1(X_0, x_0) \to \pi_1(X, x_0)\):Proof:
3.1. Let \([\alpha], [\beta] \in \pi_1(X_0, x_0)\) and suppose \(i_*([\alpha]) = i_*([\beta])\) in \(\pi_1(X, x_0)\).
3.2. By definition of path homotopy in \(X\), there exists a continuous map \(H: [0, 1] \times [0, 1] \to X\) such that:\[H(s, 0) = \alpha(s), \quad H(s, 1) = \beta(s), \quad H(0, t) = x_0, \quad H(1, t) = x_0 \quad \text{for all } s, t \in [0, 1].\]
3.3. Since the square \(I^2 = [0, 1] \times [0, 1]\) is path-connected and \(H(0, 0) = x_0\), 1 implies \(H(I^2) \subseteq X_0\).
3.4. Thus \(H\) is a continuous map into \(X_0\), which constitutes a valid basepoint-preserving path homotopy between \(\alpha\) and \(\beta\) inside \(X_0\).
3.5. Therefore \([\alpha] = [\beta]\) in \(\pi_1(X_0, x_0)\), proving that \(i_*\) is injective.
4. Conclusion:Proof:The map \(i_*: \pi_1(X_0, x_0) \to \pi_1(X, x_0)\) is a bijective group homomorphism, hence an isomorphism of groups.